Exciton Transfer Integrals Between Polymer Chains

The line-dipole approximation for the evaluation of the exciton transfer integral, $J$, between conjugated polymer chains is rigorously justified. Using this approximation, as well as the plane-wave approximation for the exciton center-of-mass wavefunction, it is shown analytically that $J \sim L$ when the chain lengths are smaller than the separation between them, or $J\sim L^{-1}$ when the chain lengths are larger than their separation, where $L$ is the polymer length. Scaling relations are also obtained numerically for the more realistic standing-wave approximation for the exciton center-of-mass wavefunction, where it is found that for chain lengths larger than their separation $J \sim L^{-1.8}$ or $J \sim L^{-2}$, for parallel or collinear chains, respectively. These results have important implications for the photo-physics of conjugated polymers and self-assembled molecular systems, as the Davydov splitting in aggregates and the F\"orster transfer rate for exciton migration decreases with chain lengths larger than their separation. This latter result has obvious deleterious consequences for the performance of polymer photovoltaic devices.